EDB — 2BP

[2BP] Prerequisites:[0KK]↺↻,[0MM]↺↻,[0K4]↺↻.Difficulty:*.Let \(Ω\) be an infinite uncountable set ; consider \(X=ℝ^Ω\) with the topology \(𝜏\) seen in [0MM]↺↻.

1. Show that every point in \((X,𝜏)\) does not admit a countable fundamental system of neighborhoods.

2. Setting

   \begin{equation} C{\stackrel{.}{=}}\{ f∈ X, f(x)≠ 0 \text{~ for at most countably many ~ } x∈Ω\} \label{eq:C} \end{equation}

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   show that \(\overline C=X\);

3. and that if \((f_ n)⊂ C\) and \(f_ n→ f\) pointwise then \(f∈ C\).

4. Let \(I\) be the set of all finite subsets of \(Ω\), this is a filtering set if sorted by inclusion; consider the net

   \[ 𝜑:I→ X\quad ,𝜑(A) = {\mathbb 1}_ A \]

   then \(∀ A∈ I, 𝜑(A)∈ C\) but

   \[ \lim _{A∈ I} 𝜑(A) = {\mathbb 1}_ X∉ C\quad . \]